Question

alexis has a total of 14 nickels and dimes worth $1.25. the system of equations {n = 14 - d, n = 25 - 2d} can be used to represent the number of nickels, n, and the number of dimes, d, alexis has. the graph of the system is shown. 3. label the axes. 4. what are any possible constraints on the x - values? 5. what are any possible constraints on the y - values? 6. what is the solution to this system of equations? 7. interpret the meaning of the solution in the context.

Explanation:

Step1: Identify variables

Let \(n\) be the number of nickels and \(d\) be the number of dimes.

Step2: Analyze constraints for \(x\) - values

The number of dimes \(d\) (assuming \(d\) is the \(x\) - variable) cannot be negative and since the total number of coins is 14, \(0\leq d\leq14\).

Step3: Analyze constraints for \(y\) - values

The number of nickels \(n\) (assuming \(n\) is the \(y\) - variable) cannot be negative and since the total number of coins is 14, \(0\leq n\leq14\).

Step4: Solve the system of equations

We have the system \(

$$\begin{cases}n = 14 - d\ =25 - 2d\end{cases}$$

\). Set the two right - hand sides equal: \(14 - d=25 - 2d\). Add \(2d\) to both sides: \(14 - d+2d=25 - 2d+2d\), which gives \(14 + d=25\). Subtract 14 from both sides: \(d = 11\). Substitute \(d = 11\) into \(n=14 - d\), we get \(n=14 - 11=3\).

Step5: Interpret the solution

The solution \((d = 11,n = 3)\) means Alexis has 11 dimes and 3 nickels.

Answer:

1. The \(x\) - axis can be labeled "Number of Dimes (\(d\))" and the \(y\) - axis can be labeled "Number of Nickels (\(n\))".
2. \(0\leq d\leq14\)
3. \(0\leq n\leq14\)
4. \((11,3)\)
5. Alexis has 11 dimes and 3 nickels.